# Maximum Log-Likelihood Estimation interpretation results

I have fitted four distributions to a sample using MLE.

The following code (example) was used to calculate the MLE in python:

from scipy.optimize import minimize
from math import exp, log

def distr(d, a):
result = (d/a**2)*exp(-d/a)
return result

def log_L(a, diameters):
result = -sum(log(distr(d, a)) for d in diameters)
return result

res = minimize(log_L, [1], args=diameters)


Which returns an output such as:

     fun: 737.6689924048228
hess_inv: array([[  5.68951613e-06]])
jac: array([[ -1.52587891e-05]])
message: Desired error not necessarily achieved due zo precision loss.
nfev: 164
nit: 7
njev: 51
status: 2
success: False
x: array([ 0.37047972])


As far as I understand it, the value "fun" of my result of the optimize.minimize function returns the actual optimized max. log-likelihood. Subsequently the values i got for "fun" for my 4 distributions:

function1 = 580.05
function2 = 1293.68
function3 = 689.63
function4 = 737.67


I'm pretty confident, that the algorithm for the MLE is correct, since I also calculated the MLE for function 4 analytically and it resulted in the identical fitted parameter.

This may now sound like a stupid question, but do I have to take the smallest or the greatest value as my best fit? I suppose its the smallest value since I minimized my log-likelihood, but I'm not completely sure.

And on the other hand, when I then want to calculate the Akaike Information Criterion (AIC), computed in the following way:

AIC = 2*k - 2* "fun"

where k is the number of parameters and "fun" is the max. log-likelihood calculated above, would I take the greatest value as my best option?

I'd appreciate any answer you could give me very much!

The formula for the AIC is: $$AIC = 2k - 2\ln(\hat{L})$$ where $\hat{L}$ is the maximum likelihood for your model. Your "fun" value corresponds to $-\ln(\hat{L})$, not $\ln(\hat{L})$. So the way you're currently computing the AIC is wrong. It should be, in your notation: AIC = 2*k + 2*"fun", i.e. '+' instead of '-' (because "fun" is already the negative log-likelihood). Note that lower AIC values are better.